The Historical Approach

By Bobby Neal Winters

Introduction: The Historical Approach

As with so many things, it came into my hands through the recommendation by a colleague of a book she had not actually read.  The book was A Radical Approach to Real Analysis by David Bressoud. The thing contained therein was the historical approach to teaching mathematics.  

Mathematicians are by their natures optimizers.  Perhaps the greatest mathematician of all time, Karl Friedrich Gauss (1777-1855), said, “A cathedral isn’t a cathedral until the last piece of scaffolding is removed.”  His personal motto was “Pauca et Matura,” few but ripe.  This is his testimony to the fact that after mathematicians have scaled to the top of the mountain with ropes and spikes and what not, they try to build trails, roads, or even rails roads up for the rest of us.

This is a great service for the rest of us, but in looking at the road, there is a tendency to forget that this is simply the endpoint of a process which as included numerous researchers, teachers, and students.

Claude Shannon (1916-2001) is known as the father of information theory.  In his theory, he imagined information going from one location to another via symbols.  The teacher/learner relationship is a system of communication, but not necessarily in the way one might naively imagine it following on the heels of that sentence. A teacher is a facilitator of the communication process. It is here that I want to thread carefully, because I dwell among those who kill upon hearing the phrase “Not a sage on the stage but a guide on the side.”

In communicating from a transmitter to a receiver, each has his (or her) on set of symbols.  Practical communication theory tells us there will probably be information lost even if both sets of symbols are the same.   In the teaching/learning process, the symbol sets are probably not the same.  The transmitter may be a German scholar who was doing his best work when Napoleon was marching across Europe and the learner might be a wannabe scholar growing up in the Oklahoma oil fields when Jimmy Carter was in the White House.

My point is there is a living system in place that transmits between those two points and more.  The Gaussian attitude of removing the scaffolding is an important part of presenting that beautiful cathedral, but we as teachers must keep in mind that the scaffolding is still out back in the shed.  Knowing that the beautification of mathematical results (or indeed the corresponding acts in any discipline) is an important part of pedagogy, should empower us as teacher-scholars.

Origins in Applications

One piece of scaffolding that often remains hidden is the Primum Movens in mathematics is physics.  I use the word physics here to be broad enough to include engineering.  This Prime Mover has interfered with mathematics multiple times in history.

The Greeks did pursue geometry for its own sake as an intellectual game, but Euclid (circa 300 BC) was known as Euclid of Alexandria.  Alexandria was the city of Alexander the Great who, while not a Greek’s Greek, did spread Greek culture to lands the old fashioned way: by conquering them.  It is not difficult to imagine Euclid and his like codifying the discoveries of those who’d been involved the many construction activities of the Egyptians. In effect, they were creating an orderly way for students to learn the geometry without to actually build the pyramid first.

Fourier Series were developed by Joseph Fourier (1768-1830) for problems connected with the theory of heat. Fourier, while getting great results, didn’t have a theoretical foundation for his mathematics.  Providing those foundations gave birth to new fields and new directions to old.

We like to teach the pursuit of knowledge for its own sake, but knowledge has an end, a purpose.  Mathematics need not teach mental masturbation. It should, in fact, teach the opposite: the delay of gratification.  The delay of gratification, I am convinced, is one of the cornerstones of civilization.

While in the process of learning to delay gratification, it is helpful to the student to know there is some gratification to be had at the end of the process.  The engineering student will endure Heat and Thermodynamics because he knows he must have it to attain his engineering credential.  He will endure Engineering Mathematics because he knows he must have it to pass Heat and Thermo.  He will endure Fourier series because he has to understand them to pass Engineering math.

As a mathematics teacher, I can help him to build that narrative. That process will be aided if I know the story myself in broad terms.  Whereas the details of the story are not a necessary part of my intellectual equipment as a mathematician, per se, they are an important part of my tool box of teacher as scholar.

Bridges

Another value to the use of the historical approach is in helping the student cross the bridge from where he is now to where he needs to be.  One commonality shared by a modern student of mathematics before taking up Fourier series and a historical figure like Fourier he began his work is an ignorance of Fourier series.  We can begin at the same place Fourier did and go from there.

Fourier didn’t spend much time on the theory.  The results he got worked very well for him when they worked. When they didn’t work, well, that could be taken care of later.  

And it was.

The first part of the mathematical program consists of calculus and other courses that are of use to engineers, physicists, chemists, biologists, and economists who are interested in mathematics as a tool rather than as an end in itself.  Mathematicians go through these ourselves because we do value the utility of the subject and want to equip ourselves in its more practical aspects; this is true, but there is more.  In addition, the number mathematics majors is typically too small to pay for a separate track for our majors.

This having been said, our majors get a somewhat skewed view of the subject.  Many absorb the view of a subject that consists of methods that have already been worked-out, methods that they need only memorize and master.  Mathematics is a living subject and our students need a bridge from the civilized, cultural center of the subject across the river of uncertainty to the frontier.

The historical approach supplies such a bridge.

As someone who has lived most of his life in the center of the country, first Oklahoma and then Kansas, I will claim some familiarity in the frontier.  Just like in the movies, there are times on the frontier where the rules are temporarily...um...ignored.  We can see Newton, Euler, and Cauchy doing things we would rap our student’s knuckles for.

Isaac Newton (1642-1727) developed series presentations for sine and cosine.  His work is absolutely brilliant, but it’s also stupid.  This is an exercise today for a freshman using Taylor’s Series.  Seeing the insight behind Taylor’s Series takes half a second, but Newton, as brilliant as he was, didn’t have it.  This didn’t stop Newton, however, as he experimented, found patterns, and verified the patterns to his satisfaction.

Leonhard Euler (1707-1783) played--and I will stick by that verb--with infinite series and obtained tremendous results.  Toward the end of his career, some of his contemporaries thought he was going off the deep end because some of his series didn’t converge.  It was more than one hundred years after his death before the foundations were laid that justified these wild calculations.

Augustin Cauchy (1789-1857) began to put analysis such as was done by Newton and Euler on firmer foundations, but in doing so he was himself faced with the dangers of virgin territory.  In Calculus I, we teach the Mean Value Theorem whose formulation is due to Cauchy.  When we prove it in the introductory analysis course, its prove can be contained in a single paragraph because the modern theory is so well developed.  Yet Cauchy’s proof is much longer and contains some jumps within it that would cause us to paint a student’s paper red: but it’s brilliant!

It is good for students to see that our subject is an adventure.  While as undergraduates, they might not have received the preparation to go into the subjects of current research interests, they have had enough to appreciate the research of the eighteenth century.  Seeing the spirit of that frontier might inspire them to explore current frontiers.

The Lasting Effect of History

As scholars who teach, we should be aware of the effects of history upon us.  We are part of a larger world that has been growing and changing for many centuries.  In many cases we bemoan the publish or perish culture that has grown up around us in modern academe, but, in doing so, we forget that publishing results is our gift to civilization.  Great minds like Archimedes (c. 287 BC – c. 212 BC) sent letters to others whom they thought might understand them.  Many times they played tricks upon their rivals to confuse them.

Others like Niccolò Fontana Tartaglia (1499/1500–1557) solved important problems like the general cubic equation, but kept the results to themselves so that they could pose challenges to others and build up their own reputations.

When Isaac Newton wrote Principia Mathematica he invented the language to describe his new system of the world, and while there might be a jot or a tittle moved here or there, we still use basically the same language today.  I am not sure he was thinking about this three hundred years ago, but the effect of his work, his words still stand.  Without learning history, however, we might not understand that he had to be pushed, bullied, and cajoled into writing his book and he made Edmund Halley pay to have it published for him.

When groups such as the Royal Society and the various journals arose as means of disseminating research results and establishing priority, progress hastened. It is difficult to over-emphasize the value of these steps toward openness and publication.  Yet, history should also show us that we are continually changing.

I believe we need to broaden our ideas of scholarship to include activies which are, in fact, necessary to teaching, learning, and research continuing in our universities.  Ernest Boyer explored this generally in his book Scholarship Reconsidered: Priorities of the Professoriate.  I believe that an exploration of the history of my subject, mathematics, vindicates this idea.  

Research and teaching are inextricably linked. Research discovers the mysteries of creation, but teaching, in a continuum of forms, transmits those discoveries to the ages.  I can have conversations with Archimedes, Newton, Euler, and Gauss because teachers who were themselves scholars have worked to make that happen.  They have learned the language of the countries of the past and have shaped the language of a boy from the Oklahoma oil field so that he may understand, if only just a little.

That is what teachers can do and that is little short of magic.